Treasures Inside the Bell: Hidden Order in Chance

TREASURES INSIDE THE BELL HIDDEN

ORDER

IN

CHANCE

C A R L O S E. P U E N T E

TREASURES HIDE Till: BELL HIDDEN ORDER IN CHANCE

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TREASURES IISIDE THE RELL H I D D E N ORDER I N C H A N C E

Carlos E. Puente Institute of Theoretical Dynamics, The University of California Davis, USA

V f e World Scientific w l

• Hong Kong New Jersey • London • Singapore Si

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Scripture texts in this work are taken from the New American Bible with Revised New Testament and Revised Psalms © 1991,1986,1970 Confraternity of Christian Doctrine, Washington, D.C. and are used by permission of the copyright owner. All Rights Reserved.

TREASURES INSIDE THE BELL Hidden Order in Chance Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-140-6 ISBN 981-238-141-4 (pbk)

Printed in Singapore.

How precious to me are your designs, 0 God; how vast the sum of them! Were I to count, they would outnumber the sands; to finish, I would need eternity. Psalm 139:17-18

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To my Father, companion and friend

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Preface The bell curve, also known as the normal or Gaussian distribution, is one of the most ubiquitous mathematical objects in science. Introduced in the early nineteenth century by mathematical prodigy Karl Friederich Gauss, the curve's fundamental nature stems from the renowned central limit theorem, first proven by Pierre-Simon Laplace, that states that a bell is the universal curve found when a multitude of unrelated (independent) quantities are added together. Recently, generalized versions of the central limit theorem leading to Gaussian distri­ butions over one and higher dimensions, via arbitrary iterations of simple mappings, have been discovered by the author and his collaborators. The purpose of this book is to reveal how these new constructions lead to infinite exotic kaleidoscopic decompositions of twodimensional circular bells in terms of beautiful deterministic patterns possessing arbitrary n-fold symmetries, patterns that, while reminding us of the infinite structure previously found in the celebrated Mandelbrot set, turn out to contain natural shapes such as snow crystals and biochemical rosettes, including life's own DNA. This book is divided into three main sections. A general introduction to the ideas is given first, followed by a gallery of patterns found inside the bell when the iterations are guided by the binary expansion of the omnipresent number ir, including a potpourri of images showing curious pattern evolutions and collages of bell patterns. Hoping to capture the most general readership, the introduction relies on several diagrams and uses as few equations as possible, leaving the technical details to a set of notes at the end of the book. This set of notes also points out relevant references, contains a sample program the reader may implement in his/her own computer in order to explore the many treasures found inside the bell, and includes the specific information for all the images in the book. To allow the reader to further appreciate the work, the book includes a CD containing selected bell patterns whose interesting evolutions may be readily visualized on a personal computer. This book could not have been accomplished without the invaluable assistance of many people. First, my warmest thanks go to Michael F. Barnsley, whose lovely ideas shaped IX

X

Preface

the direction of my research, giving birth to this work. Second, I would like to thank my collaborators throughout the years, whose dedication is reflected in the pages that follow, in particular: Enrique Angel, German Poveda, Marc Bierkens, Jorge Pinzon, Miguel Lopez, Jose Angulo, Aaron Klebanoff, John Wagner, Nelson Obregon, Demiray Simsek, Akin Orhun, Bellie Sivakumar, Stephen Bennett and Marta G. Puente. Third, I would like to acknowl­ edge the institutions that have supported my research, especially and rather appropriately Pacific Bell, whose timely donation a few years ago allowed study of the bell in peace, and the University of California, Davis where I have been given the freedom to roam beyond my natural confines as a hydrologist. I owe much to Dick Odgers, Daniel Fessler and my UCD colleagues for their trust. Fourth, my sincere thanks go to World Scientific for their professionalism in producing such a beautiful book. I am indebted to Stanley Liu and Ian Seldrup. Many thanks go also to my dear friend Fernando Duarte for his artistic cover. Finally, I would like to acknowledge the loving support of my beloved family and friends. By your constancy, you nurture and inspire my yielding to a majestic bell, one ever conducting. Carlos E. Puente Davis, California March 2002

Contents

Preface

ix

Part 1 I n t r o d u c i n g t h e Bell 1 The Notion of Iterations 2 Interpolating Wires 3 Textures on Interpolating Wires 4 The Bell as a Shadow 5 Wires and Bells in Higher Dimensions 6 Exotic Patterns Inside the Bell 7 Natural Sets Inside the Bell 8 More Treasures Inside the Bell

1 1 3 6 8 11 15 23 23

Part 2 Gallery D e s i g n s Radial Patterns Rotational Patterns Evolutions & Quilts

33 35 49 63

Part 3

73

Technical N o t e s

Index

96

xi

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Part 1

Introducing the Bell

This section of the book attempts to explain, step by step, what is needed in order to understand and recreate the lovely patterns that are found inside the bell. The material is based on a comprehensive set of diagrams and graphs, relegating the more formal details to a set of notes at the end of the book. 1

The Notion of Iterations

We begin by introducing a rather simple process named the chaos game. This is a game of chance and to play it one needs a die. To start, choose three non-aligned points and label them according to the sides of the die, 1-2, 3-4, and 5-6, as in Figure 1 (left). Then, select a point, say the one marked as 1-2, and roll the die. Suppose a 3 comes up, then move

Figure 1. First two stages of the chaos game. 1

2

Introducing the Bell

Figure 2. The chaos game after 500 (left) and 8,000 (right) tries.

from the current position (1-2) to the middle of the vertex marked with 3, namely the point shown in Figure 1 (left). Roll the die again and suppose a 5 appears. Then, move from the last point to the mid-point of the vertex given by the outcome of the die, as shown in Figure 1 (right). Repeat the process many times and see what happens. As may be hinted, the game needs to be played for a while in order to elucidate its outcome. What appears after 500 tries, and as implemented on a computer playing with fair pseudo-random numbers, may be seen in Figure 2 (left). The game ultimately generates, the celebrated Sierpinski triangle, as depicted after 8,000 rolls in Figure 2 (right). This set, known as the game's attractor, turns out to be a rather peculiar object that has a deterministic and infinite structure of "middle triangles taken away from triangles," and appears independently of the throws of the die and of the point selected to start the game. Although the formation of the triangle point by point may be surprising, its structure may be elucidated once one considers the precise rules that are being repeated, that is, iterated, again and again. If (x, y) represents a generic position, such rules are (x + x

y+ y\

Wn{x,y) = y—-—,n —^—nJ , where (xn,yn),n — 1,2,3, are the coordinates of the three vertices. Clearly, applying the mappings to the three vertices themselves gives the precedence diagram shown in Figure 3, one that depicts (using alternative line types for the three rules) the boundary conditions for the chaos game. With this pictorial aid, the deterministic structure of the Sierpinski triangle readily appears through successive iterations of the rules starting at the newly defined points and following infinite ternary trees depicting all possibilities, as in Figure 4. Notice how the triangular gaps of decreasingly small sizes in Figure 2 represent regions in space that are never visited by any of the rules. The chaos game turns out to recreate the Sierpinski triangle because following any single non-trivial branch of a ternary tree (that is, one that jumps to all the rules as guided by a fair die) rooted anywhere ultimately covers the attractor densely everywhere. This happens because the rules, by always traveling inwards, pull all successive points towards a unique and stable state. 1

3

Interpolating Wires

Figure 3. Boundary conditions for construction of the Sierpinski triangle.

Figure 4. Ternary tree depicting recursive construction of the Sierpinski triangle.

2

Interpolating Wires

As iterating a set of simple rules gives rise to interesting sets, it is relevant to study the attr actors generated by other sets of mappings. For this purpose, exchange "move to the mid-point" by the following set of two rules: wi(x,y) = ( - -x, x + di-y w2(x,y) = (- -x + - , l-x

(1) + d2-y) ,

where d\ and d2 are (free) parameters whose magnitude is less than one.

(2)

4

Introducing the Bell

Figure 5. Alternative wires interpolating {(0,0), (1/2,1), (1,0)} for sign combinations on the parameters d\ and d-2 in Equations (1) and (2). \d\\ = je^l = z. Left, z= 0.5. Right, z = 0.8.

Figure 6. Boundary conditions for construction of interpolating wires.

It happens that these simple mappings generate alternative attractors shaped as convo­ luted wires, that interpolate the points (0,0), (1/2,1) and (1,0). As portrayed in Figure 5, for alternative magnitudes and sign combinations on the parameters, such sets are functions from x to y that are either "cloud" profiles (Case + + : d\ = d2 = z) or "mountain" bound­ aries (Case H—: d\ — —d% = z; Case : —d\ = —d2 = z; and Case —\-: the reciprocal of Case -I—, not shown) that have, as the Sierpinski triangle, a rather striking repetitive nature. Notice how wires corresponding to a magnitude of z equal to 0.8 are increasingly rough and considerably longer than those for z at 0.5. As may be easily verified, Wi(0,0) = (0,0), tui(l,0) = (1/2,1), w2(0,0) = (1/2,1), and 102(1,0) = (1,0). Hence, mappings (1) and (2) yield, in parallel to Figure 3, the precedence diagram of Figure 6. Because the more general mappings may also be shown to travel inwards, they define a unique attractor which emanates from point (1/2,1) following an infinite binary tree, as depicted in Figure 7. For instance, at the first level of the construction,

Interpolating Wires

5

Figure 7. Binary tree illustrating construction of interpolating wire. Locations of nodes in the horizontal are plotted at the proper scale starting at (1/2,1), indicating that w\ travels to the left and w-i does so to the right of the mid-point.

Figure 8. Recursive construction of an interpolating wire (Case + + ) . First points are found adding z to mid-points in lines joining interpolating points. Other points are obtained adding successive powers of z from successive mid-points, as shown.

the obtained values give wi(l/2,1) = (1/4,1/2 + di), w2(l/2,1) = (3/4,1/2 + d2), that may be readily noticed in Figure 5. In the end, a chaos game dictated by fair (or biased) coin tosses may be employed to obtain the unique attractor induced by the simple mappings (1) and (2). As before, the resulting attractor turns out to be fully determined as it may be generated recursively via a geometric procedure that parallels "taking middle triangles away from triangles," as illustrated in Figure 8.2 The Sierpinski triangle and the interpolating wires (for z > 1/2) are examples of de­ terministic fractal sets defined over the plane. 3 These are objects that fill two-dimensional space to varying degrees and have fractal dimensions between 1 and 2, D — In 3/ In 2 ~ 1.585

6

Introducing the Bell

for the triangle and D = 1 + ln(2 • z)/ln(2) for the interpolating wires (that is, more as z increases, as seen in Figure 5). Notice that as z tends to its maximal value of 1, the inter­ polating wires, which are topologically one-dimensional strings, tend to fill the plane (that is, D tends to 2.)4 3

Textures on Interpolating Wires

While constructing a unique interpolating wire via the chaos game, one could also compute the relative frequencies that points make as they densely sample the attractor. Although the same set is ultimately painted via alternative iteration rules, usage of fair or biased coins yields alternative stable textures within the wire.5 For initial points equally spaced in x, as used thus far, the overall trends are as follows. When the coin is fair, a branch that goes left or right 50% of the time fills up all values within the wire evenly. This may be seen in Figure 7, by noticing the precise equally-spaced locations for branches in the x component. What is found in terms of texture is thus shown on the left in Figure 9, a uniform measure over the wire set. As displayed on the right in Figure 9, when the coin used is biased, for example, w\ used 70% of the time and w2 the remaining 30%, no longer is the uniform texture found but instead one obtains a spiky multi-layered texture, a singular multifractal measure over the wire, one that decomposes the wire into infinitely many disjoint and intertwined sets corresponding to equal (multiplicative) textures. Such a detailed and thorny texture appears because of the independence of coin tosses and may be visualized overlaying the recursive construction of such a measure in Figure 10 with the binary tree in Figure 7.6'7 Although coin tosses are employed to find a wire and its texture, the results on Figure 9 stress that chance has no influence over the objects the chaos game generates. Once the

Figure 9. Alternative textures generated on a wire via the chaos game. Fair coin (left), biased coin traveling 70% to the left and 30% to the right (right). Interpolating points: {(0,0), (1/2,1), (1,0)}, dx = 0.5, d2 = - 0 . 5 .

7

Textures on Interpolating Wires

p = 70%

q = 30%

0

(a)

1

(b)

(c)

(d)

L.LL, L L L,.. L,. ,

U„L, L,...

(e)

Figure 10. Multiplicative process defining a multifractal texture. From a uniform measure (a) to 2 1 2 = 4,096 spikes in (e). Every rectangle splits into two rectangles that have half the parent's size (horizontal length). The smaller rectangles have areas that are 70% (left) and 30% (right) of the parent's area. The first four stages are drawn to scale, but the last diagram uses a vertically-reduced scale of the true figure, which would otherwise not fit on the page.

bias or fairness of the coin is set, such a choice defines a multifractal or uniform measure that surely expresses the ultimate texture found on the same attracting wire. The peculiar nature of the measures on wires may be further explored by defining projec­ tions of wire textures along arbitrary lines. This entails moving along such a line, over the x-y plane, identifying all points within the wire that are perpendicular to such a location, and adding the textures of all those points. It turns out that such an operation is easily performed while playing the chaos game, as the sought textures are found by computing a histogram of obtained points projected over the desired line. Figure 11 illustrates these ideas when the textures in Figure 9 are projected over the x and y axes. As is seen, the obtained "shadows" along the x axis, dx, corresponding to places where the wire is intersected only once, simply give either a uniform distribution or a multifractal measure defined over the wire's domain. Along the y axis, the shadows of the wire texture, dy, become more difficult to describe as they typically add the textures found on several points, that is, two or more for the examples shown.

Introducing the Bell

8

(a)

(b)

Figure 11. Projection of a wire over x and y as given by the chaos game. Fair coin (left), biased coin traveling 70% to the left (right). Interpolating points: {(0,0), (1/2,1), (1, 0)}, dx = 0.5, d2 = - 0 . 5 . The one-dimensional histograms and wire shown approximate the true objects via 1,024 equally-spaced bins containing 50 million pseudo-random coin tosses.

As seen on the left in Figure 11, a uniform texture on such a wire results in a symmetric dy whose shape is proportional to the number of times the wire y = f(x) is crossed by horizontal lines. Observe how the wrinkles of the wire indeed reflect the shadow dy, with larger spikes corresponding to the confluence of wire crossings. As observed on the right in Figure 11, when the texture over the same wire becomes multifractal, the corresponding projection dy is no longer symmetric. This happens due to the inherent biases on the chaos game, which prefers (by layers) some points within the wire more than others. As may be appreciated, dy gives rise to an irregular and rather complex (but deterministic) object whose shape reminds us of patterns commonly observed while displaying variations of natural phenomena, either in time or in space along a given line.8 As the interpolating wires define continuous functions from x to y, the measures dx and dy shown in Figure 11 are functionally related to each other, dy is the derived measure of dx via the function / that a wire represents, or more simply, dy is found transforming dx via such a function / . 4

The Bell as a Shadow

Figure 12 shows the wires and respective projections off them over both axes (dx and dy) when the three points {(0,0), (1/2,1), (1,0)} are interpolated employing the H— rule, that is, d\ = — d2 = z in Equations (1) and (2), via a fair coin implementation of the chaos game, where the parameter z is varied from 0 up to a value of 0.999.

The Bell as a Shadow

(c)

9

(d)

Figure 12. From a uniform measure to the bell curve. Interpolating points {(0,0), (1/2,1), (1, 0)}, d\ = —d-2. = z, (a) z = 0, (b) z = 0.5, (c) z = 0.8, (d) z = 0.999. The one-dimensional histograms and wires shown approximate the true objects in 1,024 equally spaced bins containing 50 million points gathered via the chaos game. The averaged wires for high fractal dimensions have been stretched vertically for aesthetic purposes.

As may be seen, the uniform projection, always found over x, yields a variety of symmetric, derived measures over y. When z — 0, the wire simply gives linear interpolation between the three original points and hence defines a uniform measure over y. This is easily seen noticing that there are exactly two points that are crossed by an arbitrary horizontal line (except at the upper tip). When z = 0.5, the mountain-like profile gives the distribution of crossings dy already shown in Figure 11(a). As z is increased beyond 0.5, the wire becomes infinite, progressively stretches up and down, and gives shadows over y that tend to group around a central tendency. When z = 0.8, the wire attains a fractal dimension D ~ 1.678

Introducing the Bell

10

and dy exhibits two main mounds, one on each side (up-down) of the wire. When z — 0.999, the wire almost fills up two-dimensional space as D « 1.999, and dy closely approximates a bell curve.9 Even though the shown measure dy for z — 0.999 is not a true bell curve, it hints at the result that is obtained as z tends to its maximal value of 1. As such a limit is approached, the resulting wire grows to fill two-dimensional space, that is, D tends to 2, and the crossings of such a deterministic object yield a distribution that indeed tends to the bell curve.10'11

(c)

(d)

Figure 13. From a multifractal measure (70%-30% biased coin) to the bell curve. Interpolating points {(0, 0), (1/2,1), (1,0)}, di = -d2 = z, (a) z = 0, (b) z = 0.5, (c) z = 0.8, (d) z = 0.999. The one-dimensional histograms and wires shown approximate the true objects in 1,024 equally spaced bins containing 50 million points gathered via the chaos game. The averaged wires for high fractal dimensions have been stretched vertically for aesthetic purposes.

Wires and Bells in Higher Dimensions

11

Figure 13 shows the counterpart of Figure 12 when a biased coin, with a 70%-30% split, is used to define wires and textures. As seen for z = 0, a multifractal measure over x gives, as expected, another multifractal texture over y, one that is found adding the respective textures of the two horizontal crossings that the two line segments have. When z = 0.5, and as already portrayed in Figure 11(b), an asymmetric texture dy is defined which shares commonly observed erratic behavior as found on several applications. As z is increased beyond 0.5, the wires become fractal, and give derived measures dy that are progressively smooth. Surprisingly, and as found for a fair coin, the limiting derived measure dy, as z tends to 1, is also a symmetric bell curve. It happens that a bell is universally found from the same limiting wire, irrespective of the bias of the coin, and also for a vast family of textures that only excludes discrete jumps. 12 Thus, such a limiting wire perfectly filters all kinds of thorny measures and transforms them into smooth bells, whose means and standard deviations depend explicitly on the texture dx.13 Remarkably, a change of perspective by 90 degrees provides a smooth bell for a wide variety of both smooth or rough (that is, singular) measures over x and hence yields an unforeseen connection between plane-filling fractal functions and the central limit theorem.14 As turbulence phenomena are associated with multifractal behavior, plane-filling fractal interpolating functions also provide an unexpected bridge from disorder to order, one in which the opposite behaviors given by turbulence and the bell (that is, dissipation and conduction of heat) appear as two sides of the same infinite attractor. 15 5

Wires and Bells in Higher Dimensions

The simple mappings given in Equations (1) and (2) may be extended to accommodate a larger number of coordinates. For example, the two mappings: Wi(x,y,z)

= (- ■ x,H ■ x + di ■ y + hi ■ z,x + li ■ y + m,i ■ z) ,

w2(x, y,z) — ( - • x + - , —H ■ x + d2 ■ y + h2 ■ z + H, -x + l2 ■ y + m2 ■ z + 1 j , may be employed to produce a fractal interpolating function that lies in three-dimensional space, a wire, from x to the plane y-z, whose fractal dimension D could be any number in the interval [1, 3), and that interpolates the set of three points {(0, 0, 0), (1/2, H, 1), (1, 0, 0)}. 16 As illustrated in Figure 14, for a uniform texture over the shown wire, the concept of derived distributions is readily generalized in order to define projections not only over y or over z, but also over the y-z plane. These joint measures, over two dimensions, yield a wide variety of deterministic patterns whose shapes are uniquely parameterized in terms of the coefficients H,d1,d2,hi,h2,l\,l2, mi and m2, as defined in Equations (3) and (4), and the bias of the coin that allows a construction via the chaos game. As found in the one-dimensional case, it turns out that some of the shadows obtained over two dimensions closely resemble some of nature's irregular patterns, for example, weather radar images and maps of pollution concentrations. 17

(3) (4)

12

Introducing the Bell

Figure 14. A wire over three dimensions and its shadow over the plane y-z. Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, dx = d2 = 0.7, hx = h2 = -0.606, h = l2 = 0, m1 = m 2 = 0.35, fair coin.

Figure 15 shows the typical behavior obtained when the fractal dimension of the resulting wire is increased from 1 to 3. Analogous to the graphs previously shown for two-dimensional wires, each block on Figure 15 contains: the xy and xz projections of the wire itself; the implied projections over the x, y and z axes, dx, dy and dz; and a contour plot of the joint measure obtained over the plane y-z, dyz, that is, as seen from above. As is observed, an increase in fractal dimension results (as before) in progressive filtering of a multi-layered texture over x and yields joint measures that converge to a smooth bell, but now over two dimensions.18 It turns out that all interpolating wires whose fractal dimensions approach the maximal value of 3 yield joint derived measures (over y-z) that closely approximate bells over two dimensions. These space-filling objects naturally extend what was previously reported over

Wires and Bells in Higher Dimensions

13

(c)

(d)

Figure 15. From a multifractal measure (70%-30% coin bias) to a bell over two dimensions. Interpolating points {(0, 0,0), (1/2,1,1), (1, 0,0)}, dx = d2 = 0, hi = -h2 = r, h = l2 = r, m1 = m2 = 0. (a) r = 0.25, (b) r = 0.5, (c) r = 0.75 and (d) r = 0.999. All graphs summarize what is found via 15 million chaos game calculations. The averaged wires and one-dimensional histograms are based on 512 bins and the twodimensional histograms are based on 64 x 64 bins. For aesthetic purposes the averaged wires are shown vertically stretched.

two dimensions and correspond, for the two mappings in Equations (3) and (4), to 16 cases of sign combinations as follows. If the parameters for the mappings are given in polar coordinates, that is, representing a location in the plane not by their Cartesian coordinates, but using a distance from the origin and an angle from the x axis, Vf'cos^

-ri2)sin#i2)\

r[ sin d[

r\' cos d\

?2 cos#2

—r2 sin# 2

r 2 sin 02

r2 cos 92

and

I

Introducing the Bell

14

(c)

(d)

Figure 16. The geometry of correlation. Interpolating points {(0,0, 0), (1/2, H, 1), (1,0,0)}, d\ = di = 0, hl = -h2 = 0.999, h = -l2 = 0.999, mi = m 2 = 0. (a) i J = 1, (b) H = 0.41, (c) H = 0 and (d) i? = —0.41. All graphs summarize what is found via 15 million chaos game calculations. The averaged wires and one-dimensional histograms are based on 512 bins and the two-dimensional histograms are based on 64 x 64 bins. The coefficients of correlation for these bells are: (a) p = 1, (b) p = 0.7, (c) p = 0 and (d) p = —0.7. For aesthetic purposes the averaged wires are shown vertically stretched.

then joint Gaussians are found over the y-z plane whenever the magnitudes of the distances r i > r i J r2 a n d r2 (which could be either positive or negative) all tend to one, and when 0\ — d[ + &i7r and #2 — #2 + ^2^, for any k\ and k2 that are integers.19 It happens that alternative cases result in bells whose circular or elliptical nature, that is, their correlation, depends on: (a) the sign combinations on the parameters r^\ (b) the angles 6^\ (c) the coordinates of the points being interpolated, and (d) the bias of a coin used in chaos game calculations.20 To illustrate such behavior, Figure 16 shows that there are indeed space-filling wires, all based on a fair coin, that give bells with arbitrary correlations just by varying the coordinates of an interpolating point, that is, the height in y on the mid-point denoted by H. Notice how the correlation depends on the "likeness" of the xy and xz projections of the corresponding wire, that leads to the unforeseen conclusion that the correlation on two-dimensional bells is dictated by the precise geometry of space-filling wires.

Exotic Patterns Inside the Bell

6

15

Exotic Patterns Inside the Bell

Even though the extension from two-dimensional wires, that is, Equations (1) and (2), to three-dimensional wires, that is, Equations (3) and (4), is natural, the added structure on the higher dimensional mappings hampers our ability to produce an elegant general proof of the validity of the Gaussian limit.21 To this effect, we decided to study the nature of the convergence towards a two-dimensional bell to elucidate if simple trends emerged that would suggest a pathway for an alternative proof. As the two-dimensional "bells" in Figures 15 and 16 are histograms over the y and z axes and over the y-z plane of 15 million chaos game points on the limiting wire, the idea was to study how circles and ellipses would be formed when the number of points was reduced to groups of a few thousand at a time. Intuitively, we expected circles to be made of circles and disks over the y-z plane, patterns that would dance "independently" as they made-up the bell. However, we were in for a surprise, for we found that, for many combinations on the parameters r^ and 9^, such sets of a few thousand points yielded exquisite decompositions of bells in terms of crystal-like patterns that had arbitrary symmetries. It turns out that such specific patterns occur whenever all r ^ ' s tend to 1 in magnitude and when the angles on both mappings, that is, d[ , 6\' and #2 , #2 , are properly "syn­ chronized." For example, they happen for 9\ — 9X ' — 9[ and #2 = #2 = #2 when both 9\ and #2 divide 2w (360 degrees) and when such angles are multiples of one another. To illustrate the nature of these findings, Figures 17 to 20 show twenty successive sets of 10,000 points each on the y-z plane (left to right and bottom to top) that are found playing the chaos game to build a wire that interpolates the generic set of points {(0,0,0), (1/2,1,1), (1,0,0)}, such that all coefficients r^ are negative and equal, that is, (1)

(2)

(1)

(2)

n

n n r i n

r = r\' = r\' = r2 = r\ = —0.9999. Figure 17 shows what is found when the two angles 9\ and 6>2, defined above, take on the values of 120 and 60 degrees, respectively, and when a fair coin is employed in order to play the chaos game. As may be seen, with the outcomes of mappings w\ and W2 colored red and blue, the patterns exhibit a clear six-fold symmetry (360/60) and include, surprisingly and quite noticeably, shapes that resemble those present in marine microorganisms. Figure 18 is simply the counterpart of Figure 17 and shows what happens when the coin bias is changed to 30%-70%, giving a multifractal texture over the three-dimensional wire. Notice how these six-fold patterns, now more blue than red, also define beautiful shapes that include many interesting and non-trivial rosettes. Figure 19 shows what is found when the two angles 9\ and #2 take on values of 90 degrees each and when the two mappings (3) and (4) are iterated according to a fair coin. Observe now that varied beautiful patterns with four-fold symmetry (360/90) arise, shapes that ultimately coalesce to define a circular bell. Figure 20 further illustrates the trends encountered by having angles 9^ = 72 and 92 = —72 degrees. Notice how, in this case, exotic patterns having ten-fold symmetry (2 • 360/72) decompose the bell. Observe how the precise shapes from frame to frame are not easily anticipated, as they depend on the peculiar choice of heads and tails employed in selecting the two mappings Wi and W2.22

16

Introducing

the Bell

Figure 17. Sequential p a t t e r n s decomposing the two-dimensional circular bell (left to right and b o t t o m to top). Interpolating points: {(0,0, 0), ( 1 / 2 , 1 , 1 ) , (1, 0 , 0 ) } , Parameters: r[ = r[ = r2 = r2 = r = - 0 . 9 9 9 9 , #i = 120, 92 = 60, fair coin. Dots per pattern: 10,000, Scale of each box: - 9 2 . 92. Seed for pseudo random tosses: —153.

Exotic Patterns Inside the Bell

17

Figure 18. Sequential patterns decomposing the two-dimensional circular bell (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r[ ' = r[' = r 2 = r2 = r = -0.9999, 0i = 120, 02 = 60, biased coin 30-70% split, Dots per pattern: 10,000, Scale of each box: - 7 1 , 71, Seed for pseudo random tosses: —153.

18

Introducing the Bell

Figure 19. Sequential patterns decomposing the two-dimensional circular bell (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r[' = r\ ' = r2 = r2 = r = —0.9999, 61 = 90, 82 = 90, fair coin, Dots per pattern: 10,000, Scale of each box: —68, 68, Seed for pseudo random tosses: —153.

Exotic Patterns Inside the Bell

19

Figure 20. Sequential patterns decomposing the two-dimensional circular bell (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r[' = r[' = r% = »2 = r = -0.9999, 0i = 72, 02 = - 7 2 , fair coin, Dots per pattern: 10,000, Scale of each box: -169, 169, Seed for pseudo random tosses: —153.

20

Introducing the Bell

The figures just explained illustrate that the usage of chance gives, rather surprisingly, very precise shapes that define treasures inside the bell: exotic kaleidoscopes of crystal­ like symmetric patterns made explicit via the chaos game, that, when added, amazingly result in the harmonious circular (or elliptic) bell over two dimensions.23 Given the fact that a close approximation of a bell arises from a large number of mapping iterations, for example, 15 million as in Figures 15 and 16, the patterns depicted in Figures 17 to 20 represent only a tiny sample of the behavior (forever) concealed inside bivariate Gaussian distributions. 24 As may be hinted from the sequences shown, following a path of iterations, that is, a branch on the tree of Figure 7, results in an ever-changing dance of patterns, that is nicely captured by the metaphor of an aleph as described in the celebrated story by Argentine

Figure 21. Sample patterns inside the circular bell having radial symmetry. The sets have been rotated for aesthetic purposes.

Exotic Patterns Inside the Bell

21

philosopher and writer Jorge Luis Borges.25 Quite literally, choosing a set of parameters r^ whose magnitude is close to the limiting value of 1 and synchronized angles 9^ yields a "point of light" from which to see a vast number of evolving symmetric shapes. As alternative coin tosses elicit different sequences of patterns and as alternative behavior is obtained depending on the signs of the parameters r^\ the chaos game generates a vast number of "kaleidoscopic Gaussian alephs."26 In the end, and as illustrated in Figures 21 and 22 and in the accompanying CD, the circular bell contains patterns having arbitrary n-fold symmetries, sets that may be classified as having radial or rotational traits. 27 Even though pseudo-random numbers are naturally used to specify suitable iteration paths, it is relevant to emphasize that the geometric sets decomposing the bell are, in the end, deterministic designs that lie hidden inside the bell. These designs represent "alter­ native universes" that strongly depend on the actual path of iterations traveled. Invariably,

Figure 22. Sample patterns inside the circular bell possessing rotational traits.

22

Introducing the Bell

however, they provide striking crystalline kaleidoscopic mandalas having unpredictable dy­ namics, that, by varying data points, free parameters and iteration paths, define a great many nets of gems.28 Altogether, they expound gigantic jigsaw puzzles of infinite varieties, whose pieces remarkably interlock to yield bells, suggesting that inside the bell there is hidden order in chance. It is worth remarking, as already explained regarding the filtering of singular measures into bells, that it is near the Gaussian limit, when all parameters r\[> have magnitudes tending to 1, and only near such a limit, where exotic behavior is found. In fact, if these parameters are all sufficiently small (say less than 0.99 in magnitude) evolutions that parallel the ones given become rather predictable, as all sequential patterns, with say 10,000 dots, give basically the same approximation of a unique and non-crystalline attractor.

Figure 23. Sample ice crystals inside the circular bell.

Natural Sets Inside the Bell

7

23

Natural Sets Inside the Bell

Besides the mathematical relevance of the curious exotic decompositions of the bell over two dimensions, it is particularly striking to realize that some of those patterns closely resemble a host of relevant natural shapes.29 As an example, Figure 23 shows that the circular bell contains beautiful designs that correspond to the magnificent structure of the ever-changing ice crystals.30 Certainly, and more impressively, the bell also includes, as members of gigantic jigsaw puzzles, some key geometric structures of particular relevance to life itself. This is illustrated in Figure 24 that portrays: (from top to bottom) the foot-and-mouth disease virus, the E. coli chaperonin GroEl protein, the B-DNA rosette (found projecting the double helix over a plane perpendicular to its main axis), and the Salmonella typhimuriumbacteria; as found in nature (left) and as approximated inside the bell (right). 31 The patterns inside a circular bell that evoke the structure of the virus, the protein and the bacteria were chosen from pages similar to those displayed in Figures 17 to 20, but yielding the appropriate number of, in order, 5, 7 and 11 tips. Even though improved fittings may surely be found for these biochemical units, the displayed approximations should help the reader appreciate that iteration paths exist that generate via the simple mappings w± and w2 in Equations (3) and (4) (or others similar to them) objects close to those natural sets.32 As may be noticed, patterns inside the bell may be found that closely capture the very detailed structure of the ten-fold B-DNA rosette, of universal importance in the transmission of hereditary genetic information for all known life forms.33 Even though alternative iteration paths may be selected to give other close renderings of the beautiful rosette and despite the fact that a connection between the biochemistry of the DNA molecule and a geometric rosette inside the bell is yet to be established, the appearance of such a pattern indeed suggests that the Gaussian distribution may play a central role beyond the familiar fields of physics, probability and statistics. As hinted by the patterns shown thus far,34 the bell represents an unexpected repository of shapes that, by providing suitable blueprints of key building blocks in physics and biology,35 establishes a new paradigm for addressing the very relevant questions pertaining to the origins of order, one containing a rather minimal computational complexity that may be coined "order at the plenitude of dimension." m 8

More Treasures Inside the Bell

While one ponders the equation that defines the beautiful standard bell curve

fix) = - L e~x2/2> one readily recognizes three key irrational numbers that help us understand generic shapes in our daily life: squares in y/2, spirals in e, and circles in w. As these numbers generate, in their binary expansions, sequences of digits that appear to be guided by "chance," (that is, \[2 = 1.0110100..., e = 10.101101... ,TT = 11.001001...) it becomes natural to ask

24

Introducing the Bell

Figure 24. Biochemical patterns inside the circular bell.

More Treasures Inside the Bell

25

Figure 25. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of \/2 (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r\ ' = rf] = r^] = r f ' = r = -0.99999999, 0i = 0, 92 = 30, Dots per pattern: 25,000, Scale of each box: -173, 173.

if they encode, via iterations of the simple mappings (3) and (4), relevant patterns inside two-dimensional bells. In this spirit, Figures 25 to 28 present some examples of what the bell contains when such numbers set up a dialogue between mappings w\ and u>2,foralternative space-filling wires passing by the generic interpolating points, that is, {(0, 0,0), (1/2,1,1), (1, 0,0)}, that result in, 12-, 8-, 6- and 10-fold symmetry, in that order. As seen in these figures, the digits of these irrational numbers (and surely others) do provide beautiful decompositions of the circular bell. Notice by comparing Figure 17 with Figure 27 that usage of the "random" digits of n yields other exotic patterns that, although resembling the ones reported via pseudo-random numbers, are (as expected) quite distinct. As seen in Figure 28, and via enlargements in Figures 29 and 30, the digits of % sur­ prisingly abridge relevant natural patterns, as the second set, made up of 20,000 dots, gives

26

Introducing the Bell

Figure 26. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of e (left to right and bottom to top). Interpolating points: {(0,0, 0), (1/2,1,1), (1,0,0)}, Parameters: —r[ — r\ ' = { ] rW = r 2 =r = 0.99999999, 6>i = 0, 92 = 45, Dots per pattern: 25,000, Scale of each box: -479, 479.

a topologically correct approximation to the projection of our B-DNA, as already seen in Figure 24. Notice the appearance of rings and spokes in the real and generated patterns that leads us to an unforeseen connection between the geometric structure of life (albeit frozen over two dimensions) and the binary digits of n through the bell.37 This intriguing linkage insinuates an unexpected avenue for finding "meaning" in the intrinsic "randomness" in 7r, and leads us to wonder what else could its binary expansion (or others) encode inside the bell (or others) in two and in higher dimensions. A preliminary analysis of the first 100,000 binary digits of y/2 and e reveals that they do not encode via the generic interpolating points, either every 10,000 or 20,000 dots, the B-DNA rosette. However, these results do not preclude the existence of such a shape later on or via alternative wires passing by other points and rather invite us to further study the mysteries of arbitrary symmetries that are encoded in the bell via such key numbers (and

More Treasures Inside the Bell

27

Figure 27. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of i: (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: r\ = r[ = 2) =r = -0.9999, 6»i = 120, 62 = 60, Dots per pattern: 10,000, Scale of each box: - 9 5 , 95. r W = r2

28

Introducing the Bell

Figure 28. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of 7r (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: —r[ = r\ = r(21] = rf] =r = 0.99999999, 61 = 36, 02 = 36, Dots per pattern: 20,000, Scale of each box: -556, 556.

More Treasures Inside the Bell

29

Figure 29. Sequential patterns decomposing the two-dimensional circular bell via the binary digits of n (left to right and bottom to top). Interpolating points: {(0,0,0), (1/2,1,1), (1,0,0)}, Parameters: —r\ — r[ — r{21] = r{22) =r = 0.99999999, 9X = 36,